A binary code is said to be a disjunctive (s, l) cover-free code if it is an incidence matrix of a family of sets where the intersection of any l sets is not covered by the union of any other s sets of this family. A binary code is said to be a list-decoding disjunctive of strength s with list size L if it is an incidence matrix of a family of sets where the union of any s sets can cover no more that L ? 1 other sets of this family. For L = l = 1, both definitions coincide, and the corresponding binary code is called a disjunctive s-code. This paper is aimed at improving previously known and obtaining new bounds on the rate of these codes. The most interesting of the new results is a lower bound on the rate of disjunctive (s, l) cover-free codes obtained by random coding over the ensemble of binary constant-weight codes; its ratio to the best known upper bound converges as s ? ?, with an arbitrary fixed l ? 1, to the limit 2e?2 = 0.271 ... In the classical case of l = 1, this means that the upper bound on the rate of disjunctive s-codes constructed in 1982 by D'yachkov and Rykov is asymptotically attained up to a constant factor a, 2e?2 ≤ a ≤ 1.

Bounds on the rate of disjunctive codes

A binary code is called a superimposed cover-free (s, ℓ)-code if the code is identified by the incidence matrix of a family of finite sets in which no intersection of ℓ sets is covered by the union of s others. A binary code is called a superimposed list-decoding sL-code if the code is identified by the incidence matrix of a family of finite sets in which the union of any s sets can cover not more than L - 1 other sets of the family. For L = ℓ = 1, both of the definitions coincide and the corresponding binary code is called a superimposed s-code. Our aim is to obtain new lower and upper bounds on the rate of the given codes. The most interesting result is a lower bound on the rate of superimposed cover-free (s, ℓ)-codes based on the ensemble of constant weight binary codes. If the parameter ℓ ≥ 1 is fixed and s → ∞, then the ratio of this lower bound to the best known upper bound converges to the limit 2 e-2 = 0.271. For the classical case ℓ = 1 and s ≥ 2, the given statement means that the upper bound on the rate of superimposed s-codes obtained by A.G. Dyachkov and V.V. Rykov (1982) is asymptotically attained to within a constant factor a, 2 e-2 ≤ a ≤ 1.

Bounds on the rate of superimposed codes

An $(r,w;d)$ cover-free family $(CFF)$ is a family of subsets of a finite set such that the intersection of any $r$ members of the family contains at least $d$ elements that are not in the union of any other $w$ members. The minimum number of elements for which there exists an $(r,w;d)-CFF$ with $t$ blocks is denoted by $N((r,w;d),t)$. 
In this paper, we show that the value of $N((r,w;d),t)$ is equal to the $d$-biclique covering number of the bipartite graph $I_t(r,w)$ whose vertices are all $w$- and $r$-subsets of a $t$-element set, where a $w$-subset is adjacent to an $r$-subset if their intersection is empty. Next, we introduce some new bounds for $N((r,w;d),t)$. For instance, we show that for $r\geq w$ and $r\geq 2$ 
$$ N((r,w;1),t) \geq c{{r+w\choose w+1}+{r+w-1 \choose w+1}+ 3 {r+w-4 \choose w-2} \over \log r} \log (t-w+1),$$ where $c$ is a constant satisfies the well-known bound $N((r,1;1),t)\geq c\frac{r^2}{\log r}\log t$. Also, we determine the exact value of $N((r,w;d),t)$ for some values of $d$. Finally, we show that $N((1,1;d),4d-1)=4d-1$ whenever there exists a Hadamard matrix of order 4d.

Some New Bounds For Cover-Free Families Through Biclique Cover

Some new bounds for cover-free families through biclique covers

A (w,r) cover-free family is a family of subsets of a finite set such that no intersection of w members of the family is covered by a union of r others. A (w,r) superimposed code is the incidence matrix of such a family. Such a family also arises in cryptography as the concept of key distribution pattern. In the present paper, we give some new results on superimposed codes. First we construct superimposed codes from super-simple designs which give us results better than superimposed codes constructed by other known methods. Next we prove the uniqueness of the (1,2) superimposed code of size 9 × 12, the (2,2) superimposed code of size 14 × 8, and the (2,3) superimposed code of size 30 × 10. Finally, we improve numerical values of upper bounds for the asymptotic rate of some (w,r) superimposed codes. © 2004 Wiley Periodicals, Inc.

Some new results on superimposed codes

A binary code is called a (w, r) cover-free code if it is the incidence matrix of a family of sets where the intersection of any w of the sets is not covered by the union of any other r sets. Such a family is called a (w, r) cover-free family. We obtain a new recurrent inequality for the rate of (w, r) cover-free codes, which improves previously known upper bounds on the rate.

Asymptotic Upper Bound for the Rate of ( w , r ) Cover-Free Codes

A (w, r)-cover-free code is the incidence matrix of a family of sets where no intersection of w members of the family is covered by the union of r others. We obtain a new condition in view of which (w, r)-cover-free codes with a simple structure are optimal. We also introduce (w, r)-cover-free codes with a constraint set.

On the Optimality of Trivial ( w , r )-Cover-Free Codes

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Asymptotic Upper Bound for the Rate of ( w , r ) Cover-Free Codes

On the Optimality of Trivial ( w , r )-Cover-Free Codes